Topological study of persistent homology on complicated force chain network evolution in granular media
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摘要: 力鏈網絡是表征顆粒物質和散體介質細觀力學本質特征以及研究其宏觀結構力學行為的重要基礎。由于力鏈構型的復雜性和多重形態，使得力鏈網絡難以分析處理。拓撲數據分析（TDA）為定量描述力鏈網絡提供了一種簡單、有效且易處理的方法。基于拓撲數據分析中的持續同調理論，建立了散體顆粒從接觸網絡到力鏈網絡，再到拓撲空間模型的分析方法。以放頂煤開采為例，采用光彈實驗方法，追蹤放頂煤開采力鏈演化過程，對頂煤與覆巖力鏈網絡進行持續同調拓撲分析。研究表明：將采場中散體顆粒應力作為持續同調分析的閾值ε時，Betti數可以作為衡量采場來壓的一個量化指標，采場頂部覆巖和工作面前方煤巖力鏈網絡β0曲線呈拋物線形，且來壓時拋物線峰值高于初始狀態下的峰值。頂部覆巖β0峰值位于強力鏈范圍，而工作面前方煤巖β0峰值位于次強力鏈范圍。β1曲線呈L形，在[0.3, 1]區間內，β1趨近于0，在[0, 0.2]區間內與ε近似呈反比。持續同調拓撲研究為定量分析顆粒介質復雜力鏈網絡演化和宏觀力學行為提供了有效方法。Abstract: The force chain network is a crucial basis for characterizing the essential features of granular media microscopic mechanics and for studying the mechanical behavior of their macroscopic structures. Due to the complexity and multiformity of the force chain configuration, the topological data analysis (TDA) method provides a simple, effective, and manageable method to quantitatively describe the force network. Based on the theory of persistent homology in TDA, the analysis method of granules from contact network to force network to topological model is established. The two key factors affecting the structure of a force chain network are its connectivity and closure. β0 reflects the number of particle clusters whose normal force is greater than the provided threshold. β1 reflects the number of holes in the force chain cluster. Persistent homology, a mathematical method to calculate the topological features of structures in metric spaces with different resolutions, is applied to the structural analysis of force chain networks. Different from other methods that separately consider force threshold levels, the evolution of a force chain structure at the force threshold level is helpful in understanding the persistence characteristics of geometric structures at various force levels and describing the force chain network completely and deeply. For example, the top coal and overburden force chain evolution in top-coal caving mining is investigated through photoelastic experiments, and the TDA of the top-coal caving force network chain is performed. The research shows that when the particle stress is obtained as the threshold ε of persistent homology analysis in top-coal caving mining, the Betti number in persistent homology analysis can be used to evaluate periodic weighting. In the overlying strata and front of the working face, the curves of β0 are parabolic, of which the peak values under the periodic weighting are higher than that under the initial condition. Furthermore, the difference between them is that the peak values in the overlying strata are located in the range of strong force chains, whereas those in front of the working face are located in the range of sub-strong force chains. The curves of β1 show L-shaped, suggesting that within [0.3, 1], β1 approaches 0; within [0, 0.2], β1 is approximately inversely proportional to ε; and within [0.3, 1], β1 trends to 0. Topological studies of persistent homology provide an effective method for quantitative analysis of the complicated force chain network evolution and macroscopic mechanical behavior in granular media.
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圖 2 力鏈網絡. (a) 顆粒體系；（b）閾值為1.4倍平均應力時的力鏈網絡；（c）閾值為0.5倍平均應力時的力鏈網絡；（d）閾值為0時的力鏈網絡
Figure 2. Force chain network: (a) particle system; (b) force chain network with the threshold value of 1.4 times the average stress; (c) force chain network with the threshold value of 0.5 times the average stress; (d) force chain network with the threshold value of 0
圖 4 光彈實驗模型(單位：mm): 1—煤層; 2—基本頂; 3—上部覆巖
Figure 4. Photoelastic experiment model (unit: mm): 1—coal seam; 2—main roof; 3—overlying strata
圖 5 放頂煤開采光彈力鏈網絡. （a）工況0（初始狀態）；（b）工況3；（c）工況5；（d）工況8
Figure 5. Photoelastic images of force chains in top coal caving mining: (a) initial state; (b) mining step 3; (c) mining step 5; (d) mining step 8
圖 6 不同工況下的力鏈平均強度
Figure 6. Average strength of force chains under different mining steps
圖 10 頂部覆巖不同工況下的Betti數隨閾值ε變化曲線. (a) β0; (b) β1
Figure 10. Curves of Betti values vs ε under different mining steps in overburden: (a) β0; (b) β1
久色视频
圖 11 工作面前方煤巖不同工況下的Betti數隨閾值ε變化曲線. (a) β0; (b) β1
Figure 11. Curves of Betti values vs ε under different mining steps in front of working face: (a) β0; (b) β1.
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